If you need to delete exactly "Recent Searches" - no need to delete all cookies and caches as suggested!
In 2021, recent searches are stored in the "Local Storage" of the browser under the key "JobSearchUI:recentSearches".
In Chrome, you can open the DevTools (press F12), go to the Application tab (may be hidden under ">>" option) and select Local Storage of the "
upwork.com" there.
Then, manually or using the Filter, find an entry with the "JobSearchUI: recentSearches" key and delete it via the context menu that opens with the right mouse button.
Now you can close the DevTools by pressing F12 again and refresh the page - the list of recent searches will be empty.